1. Field of the Invention
The present invention relates to Sigma-Delta Digital-to-Analog (D/A) and Analog-to-Digital (A/D) converters. More particularly, the present invention relates to low-cost methods to compensate for quadratic and cubic errors in Sigma-Delta D/A and A/D converters.
2. The Prior Art
The Sigma-Delta modulator forms the basis for many modern Analog-to-Digital (A/D) and Digital-to-Analog (D/A) converter topologies. In one fairly common implementation of a D/A converter, a digital Sigma-Delta modulator takes in a relatively high-resolution but highly over-sampled digital input signal and converts it to a one-bit highly over-sampled digital output signal. The pulse density, or duty factor, of the one-bit output represents the signal. This one-bit signal is fed to a one-bit D/A converter which, by virtue of its simplicity, is generally lower in cost than a higher-resolution D/A converter.
Because the resolution of the digital signal is drastically lowered by the sigma-delta modulator, a large amount of quantization noise is added to the signal. However, due to the well-known frequency-shaping operation of the sigma-delta modulator, the quantization noise is added largely in frequency bands not occupied by the desired signal, and can thus be filtered away, leaving just the desired analog signal at the output of the converter.
The performance of the Sigma-Delta converter can be similar to that of a much higher resolution conventional converter. For example, a Sigma-Delta D/A converter comprising a one-bit modulator and a one-bit D/A and an output filter can have similar performance in many respects to a 16-bit conventional (e.g., R-2R ladder) D/A converter.
In a one-bit Sigma-Delta A/D converter, the Sigma-Delta modulator is mostly analog. An analog-input comparator and a one-bit D/A converter in the feedback loop of the modulator provide the interface between the analog and digital domains.
Practical Sigma-Delta D/A and A/D converters may exhibit Integral Non-Linearity (INL) errors for many reasons. One such reason includes errors due to Inter-Symbol Interference (ISI). This occurs where like digital symbols do not have a like effect in the analog domain because symbols are not completely independent of each other, and are influenced by the history of other recent symbols. For example, the DC result of converting the one-bit sequence of digital symbols “010010” to analog may be slightly different than the result from the sequence “001100,” even though the average pulse density for the two sequences is the same, because in one case the two “1” bits are back-to-back and in the other case they are not. Often, this difference will exhibit itself primarily as a second-order (or quadratic) non-linearity of the D/A (or A/D) output.
Besides ISI, there are many other possible root-causes of INL including, for example, inherent component effects, such as the resistance sensitivity to an applied voltage of a resistor. When a sine wave is applied to a converter with INL, harmonic distortion products result.
Previous designs to compensate quadratic or cubic non-linearities generally required the use of multipliers either in the analog or digital domain. For instance, the input number to a Sigma-Delta D/A converter might be multiplied by itself (i.e., squared) and then multiplied by a compensation coefficient, and then added to the original input, in order to pre-compensate a non-linearity that occurs in the analog portion of the D/A converter (e.g., due to ISI). In this case, two multiply operations were required. Generally for a cubic compensation, three multipliers would be required, and for simultaneous quadratic and cubic compensation, a total of four multipliers would be used. Such an arrangement is shown in FIG. 1. Multipliers, whether implemented in the analog or digital domain, are relatively expensive components.